\(\begin{cases} \frac{-c}{b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - c \cdot \left(a \cdot 4\right)}}{2 \cdot a} & \text{otherwise} \end{cases}\)
- Started with
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
30.5
- Applied taylor to get
\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
24.5
- Taylor expanded around inf to get
\[\begin{cases} \frac{2 \cdot c}{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
24.5
- Applied simplify to get
\[\color{red}{\begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}}\]
24.5
- Using strategy
rm 24.5
- Applied add-exp-log to get
\[\begin{cases} \color{red}{\frac{c}{\frac{c}{b} \cdot a - b}} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{e^{\log \left(\frac{c}{\frac{c}{b} \cdot a - b}\right)}} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
33.5
- Applied taylor to get
\[\begin{cases} e^{\log \left(\frac{c}{\frac{c}{b} \cdot a - b}\right)} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} e^{\left(\log c + \log -1\right) - \log b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
45.4
- Taylor expanded around 0 to get
\[\begin{cases} e^{\color{red}{\left(\log c + \log -1\right) - \log b}} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} e^{\color{blue}{\left(\log c + \log -1\right) - \log b}} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
45.4
- Applied simplify to get
\[\begin{cases} e^{\left(\log c + \log -1\right) - \log b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} c \cdot \frac{-1}{b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{otherwise} \end{cases}\]
9.1
- Applied final simplification
- Applied simplify to get
\[\color{red}{\begin{cases} c \cdot \frac{-1}{b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{-c}{b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - c \cdot \left(a \cdot 4\right)}}{2 \cdot a} & \text{otherwise} \end{cases}}\]
9.0
